A discrete conservative model for the linear vibrating string and rod

In this paper, we shall suggest and study a conservative discrete model for the linear vibrating string and rod fixed at the end points. We shall prove that the difference systems involved in our models may be seen as second-order unconditionally stable finite difference schemes of the classical equations of the linear vibrating string and vibrating rod. If the forces acting on the string (or rod) are conservative the total energy of the discrete solutions of our models is conserved and we can prove that we have stability for every choice of the time step Δt. We have considered both hinged and clamped rod; the constrains are naturally included into the model and the conservation of energy is still proved by giving a suitable definition of potential energy. Some numercial examples are presented.